Definition of elves

An elf is a scale in a regular temperament which is tempered from a JI scale in the group of the temperament which is epimorphic via a val V which may not be, and characteristically is not, a val supporting the temperament. This allows the elf to have more freedom in scale size and structure but still to possess the coherence induced by the epimorphic mapping.

To construct an elf, take the intervals in the JI group of the temperament which lie within an octave and keep only the least complex (in terms of Benedetti height) representative for each corresponding interval of the temperament. Order the remaining JI intervals by increasing temperamental complexity, breaking ties by increasing Benedetti complexity. For each integer value 1 ≤ i ≤ V(2), set the ith element of a transversal for the scale to be the first interval c in the listing such that V(c) = i; which is to say, the interval of least temperamental complexity with ties broken by Benedetti height. The tempering of this transversal by a tuning map for the temperament is the elf.

Rank two examples

13-limit leapday

elfleapday7
elfleapday8d
elfleapday9
elfleapday10
elfleapday12f

11-limit magic

elfmagic7
elfmagic8
elfmagic8d
elfmagic9
elfmagic10
elfmagic12

11-limit miracle

elfmiracle7
elfmiracle8d
elfmiracle9
elfmiracle10
elfmiracle12

13-limit myna

elfmyna7
elfmyna8d
elfmyna9
elfmyna10
elfmyna12f

13-limit octacot

elfoctacot7
elfoctacot8d
elfoctacot9
elfoctacot10
elfoctacot12f

13-limit qilin

elfqilin7
elfqilin8d
elfqilin9
elfqilin10
elfqilin12f

13-limit sensus

elfsensus7
elfsensus8d
elfsensus9
elfsensus10
elfsensus12
elfsensus12f

11-limit valentine

elfvalentine7
elfvalentine8d
elfvalentine9
elfvalentine10
elfvalentine12

Rank three examples

11-limit jove

elfjove7
elfjove8d
elfjove9
elfjove10
elfjove11c
elfjove12

13-limit madagascar

elfmadagascar7
elfmadagascar8d
elfmadagascar9
elfmadagascar10
elfmadagascar12f
elfmadagascar14c
elfmadagascar15

11-limit portent

elfportent9
elfportent10
elfportent11c
elfportent12
elfportent15

11-limit thrush

elfthrush7
elfthrush8d
elfthrush9
elfthrush10
elfthrush12

11-limit zeus

zeus7tri
elfzeus8
elfzeus9
elfzeus10
elfzeus12

Rank four examples

Keenanismic

elfkeenanismic7
elfkeenanismic8d
elfkeenanismic9
elfkeenanismic10
elfkeenanismic11c
elfkeenanismic12
elfkeenanismic19

Swetismic

elfswetismic8d
elfswetismic9
elfswetismic10
elfswetismic12

Valinorsmic

elfvalinorsmic7
elfvalinorsmic8d
elfvalinorsmic9
elfvalinorsmic10
elfvalinorsmic11c
elfvalinorsmic12